*4.2. Carbon Flux Modeling for the Carburization Process*

The amount of carbon atoms transferred at the metal surface is defined as carbon flux, which is determined by temperature and pressure, according to physicochemical principles. In practical, the carbon transfer from the atmosphere to the steel surface is proportional to the adsorption capacity of carburizing gas medium, so it follows the empirically derived Freundlich equation [25]:

$$J = k p^{\frac{kT}{A}} \tag{3}$$

As can be seen from the above analysis, in the first carburizing pulse, which occurs generally within the first 30 s, the carbon concentration gradient and the carbon flux are both relatively large. When the carbon concentration on the workpiece surface reaches a certain level (an intermediate carbon concentration), the carbon concentration gradient decreases by more than 80%. The driving force for carbon atoms to enter the sample has decreased by a relatively large amount, and the carbon flux also shows an order-of-magnitude decrease, which is a process maintained in the subsequent 30 s to 90 s. After this, the carbon flux value of the pulse takes the average carbon flux value at 90 s owing to its small carbon concentration gradient. In the carburization process, the initial carburization of the first pulse plays a dominant role in carburization. Therefore, a regression analysis on the adsorption amount of three materials under different carburization temperature and pressure conditions within the first 30 s was conducted. The carbon flux values of the three materials in 30 s are shown in Tables 2–4.


**Table 2.** Effect of carburizing temperature and pressure on average carbon flux of 12Cr2Ni4A steel.

**Table 3.** Effect of carburizing temperature and pressure on average carbon flux of 16Cr3NiWMoVNbE steel.


**Table 4.** Effect of carburizing temperature and pressure on average carbon flux of 18Cr2Ni4WA steel.


Equation (3) is converted to obtain Equation (4):

$$
\ln l = \ln k + \frac{RT}{A} \ln p = \frac{R}{A} \ln p \cdot T + \ln k\_r \tag{4}
$$

$$\frac{R}{A}\ln p = M\_\prime \tag{5}$$

$$
\ln k = N.\tag{6}
$$

*p* is taken as a constant. In this way, linear regression can be performed on the data of the three different materials in Tables 2–4.

Figure 15 shows the regression curves of carburization temperature at different pressure.

**Figure 15.** Regression curves of carburization temperature and ln*J*: (**a**) 12Cr2Ni4A steel, (**b**) 16Cr3NiW MoVNbE steel, and (**c**) 18Cr2Ni4WA steel.

The regression results are shown in Equations (7)–(9). For 12Cr2Ni4A steel:

$$J = 3.26 \times 10^{-9} \cdot p^{\frac{8.3447}{4412}}.\tag{7}$$

For 16Cr3NiWMoVNbE steel:

$$J = 1.14 \times 10^{-8} \cdot p^{\frac{8.3147}{5512}}.\tag{8}$$

For 18Cr2Ni4WA steel:

$$J = 2.35 \times 10^{-10} \cdot p^{\frac{8.3447}{9988}}.\tag{9}$$
